6.1 Exponential functions  (Page 11/16)

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 $x$ 1 2 3 4 $f\left(x\right)$ 70 40 10 -20

Linear

 $x$ 1 2 3 4 $h\left(x\right)$ 70 49 34.3 24.01
 $x$ 1 2 3 4 $m\left(x\right)$ 80 61 42.9 25.61

Neither

 $x$ 1 2 3 4 $f\left(x\right)$ 10 20 40 80
 $x$ 1 2 3 4 $g\left(x\right)$ -3.25 2 7.25 12.5

Linear

For the following exercises, use the compound interest formula, $\text{\hspace{0.17em}}A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}.$

After a certain number of years, the value of an investment account is represented by the equation $\text{\hspace{0.17em}}10,250{\left(1+\frac{0.04}{12}\right)}^{120}.\text{\hspace{0.17em}}$ What is the value of the account?

What was the initial deposit made to the account in the previous exercise?

$10,250$

How many years had the account from the previous exercise been accumulating interest?

An account is opened with an initial deposit of $6,500 and earns $\text{\hspace{0.17em}}3.6%\text{\hspace{0.17em}}$ interest compounded semi-annually. What will the account be worth in $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ years? $13,268.58$ How much more would the account in the previous exercise have been worth if the interest were compounding weekly? Solve the compound interest formula for the principal, $\text{\hspace{0.17em}}P$ . $P=A\left(t\right)\cdot {\left(1+\frac{r}{n}\right)}^{-nt}$ Use the formula found in the previous exercise to calculate the initial deposit of an account that is worth $\text{\hspace{0.17em}}14,472.74\text{\hspace{0.17em}}$ after earning $\text{\hspace{0.17em}}5.5%\text{\hspace{0.17em}}$ interest compounded monthly for $\text{\hspace{0.17em}}5\text{\hspace{0.17em}}$ years. (Round to the nearest dollar.) How much more would the account in the previous two exercises be worth if it were earning interest for $\text{\hspace{0.17em}}5\text{\hspace{0.17em}}$ more years? $4,572.56$ Use properties of rational exponents to solve the compound interest formula for the interest rate, $\text{\hspace{0.17em}}r.$ Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of$9,000 and was worth $13,373.53 after 10 years. $4%$ Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded monthly, had an initial deposit of$5,500, and was worth \$38,455 after 30 years.

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.

$y=3742{\left(e\right)}^{0.75t}$

continuous growth; the growth rate is greater than $\text{\hspace{0.17em}}0.$

$y=150{\left(e\right)}^{\frac{3.25}{t}}$

$y=2.25{\left(e\right)}^{-2t}$

continuous decay; the growth rate is less than $\text{\hspace{0.17em}}0.$

Suppose an investment account is opened with an initial deposit of $\text{\hspace{0.17em}}12,000\text{\hspace{0.17em}}$ earning $\text{\hspace{0.17em}}7.2%\text{\hspace{0.17em}}$ interest compounded continuously. How much will the account be worth after $\text{\hspace{0.17em}}30\text{\hspace{0.17em}}$ years?

How much less would the account from Exercise 42 be worth after $\text{\hspace{0.17em}}30\text{\hspace{0.17em}}$ years if it were compounded monthly instead?

$669.42$

Numeric

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.

$f\left(x\right)=2{\left(5\right)}^{x},$ for $\text{\hspace{0.17em}}f\left(-3\right)$

$f\left(x\right)=-{4}^{2x+3},$ for $\text{\hspace{0.17em}}f\left(-1\right)$

$f\left(-1\right)=-4$

$f\left(x\right)={e}^{x},$ for $\text{\hspace{0.17em}}f\left(3\right)$

$f\left(x\right)=-2{e}^{x-1},$ for $\text{\hspace{0.17em}}f\left(-1\right)$

$f\left(-1\right)\approx -0.2707$

$f\left(x\right)=2.7{\left(4\right)}^{-x+1}+1.5,$ for $f\left(-2\right)$

$f\left(x\right)=1.2{e}^{2x}-0.3,$ for $\text{\hspace{0.17em}}f\left(3\right)$

$f\left(3\right)\approx 483.8146$

$f\left(x\right)=-\frac{3}{2}{\left(3\right)}^{-x}+\frac{3}{2},$ for $\text{\hspace{0.17em}}f\left(2\right)$

Technology

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.

$\left(0,3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,375\right)$

$y=3\cdot {5}^{x}$

$\left(3,222.62\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(10,77.456\right)$

$\left(20,29.495\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(150,730.89\right)$

$y\approx 18\cdot {1.025}^{x}$

$\left(5,2.909\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(13,0.005\right)$

$\left(11,310.035\right)\text{\hspace{0.17em}}$ and $\left(25,356.3652\right)$

$y\approx 0.2\cdot {1.95}^{x}$

Extensions

The annual percentage yield (APY) of an investment account is a representation of the actual interest rate earned on a compounding account. It is based on a compounding period of one year. Show that the APY of an account that compounds monthly can be found with the formula $\text{\hspace{0.17em}}\text{APY}={\left(1+\frac{r}{12}\right)}^{12}-1.$

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